Kajal holds a ruler in some wavy water. The depth of the water $t$ seconds after she starts measuring it, in $\text{cm}$, is given by $ D(t) = 50 - 23 \sin \left(\pi (t+0.23)\right)$. After she starts measuring, when is the first time the depth of the waves is at its average value? Give an exact answer. When $t =~$
Solution: First, let's look at when $\sin u$ achieves its average value of $0$. $\sin u = 0$ when $u$ is a multiple of $\pi$. Since $\sin u$ achieves its average value when $u$ is a multiple of $\pi$, $D(t)$ achieves its average value when $\pi(t +0.23) = \pi n$ for $n$, an integer. That happens when $t + 0.23$ is an integer, or when $t = -0.23, 0.77, 1.77, 2.77,$ etc. The first positive value where the depth is at its average value is $0.77$ seconds after Kajal starts measuring.